# Best Representations of 3-Dimensional Symmetry: Regular Polygon Faces

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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## What Do We Mean By "Best?"

I was first inspired to consider this problem while building a set of solids to represent all 32 crystal classes (most commercial model sets omit the less common and less symmetrical classes). I soon realized the answers were different depending on whether I was cutting models out of wood or building them out of cardboard, and whether I was thinking of ease of construction or aesthetic appeal.

Some reasonable possible definitions of "best" include:

 1 1*

## Monoclinic

 2 m Augmented Sphenocorona (J87) 17 faces 2/m

## Orthorhombic

 222 mm Augmented triangular prism (J49) 8 faces 2/m 2/m 2/m Bilunabirotunda (J91) 14 faces

## Uniaxial Classes

The uniaxial classes which have a single major symmetry axis and additional twofold axes or mirror planes all have certain features in common. For each group, whether trigonal, tetragonal or hexagonal, there are seven possible classes (but some turn out to be degenerate). These can all be derived by taking one of the seven strip space groups and wrapping it around a cylinder. If N is the degree of symmetry, we have:

N
A single major symmetry axis alone
N/m
Symmetry axis perpendicular to a mirror plane
Nm
Symmetry axis with mirror planes intersecting along the axis
N/m m
Symmetry axis perpendicular to a mirror plane, and mirror planes intersecting along the axis. A combination of N/m and Nm. This is the holosymmetric class: it contains all the other symmetries as subsets
N2
Symmetry axis with 2-fold axes perpendicular to it
N*
N-fold rotoinversion axis
N*2m
N-fold rotoinversion axis with 2-fold axes perpendicular to it and with mirror planes intersecting along the N-fold axis

## Trigonal - Rhombohedral

 3 3m Elongated trigonal prism 7 faces (Same as 6*) 3/m (Same as 6* 2/m) 3/m m 32 Gyroelongated Triangular Bicupola (J44) 26 faces 3* 3*2m Elongated Triangular Gyrobicupola (J36). Eliminating the central prism results in a cuboctahedron. 20 faces

## Tetragonal

 4 4/m 4mm Elongated square pyramid. Also regular faced. 9 faces 4/m 2/m 2/m Elongated octahedron. Also regular faced. 12 faces 422 Gyroelongated square bicupola (J45) 34 faces 4* 4* 2/m Gyrobifastigium. Also regular faced. 8 faces

## Hexagonal

 6 6/m 6mm 6/m 2/m 2/m Hexagonal Prism (equilateral) 8 faces 622 6* 6* 2/m Trigonal prism. Also minimal  5 faces

## Isometric

 4/m 3* 2/m Cube. Also isohedral, minimal and regular-faced. Other equilateral examples are the octahedron and rhombic dodecahedron, and all Archimedean polyhedra with cubic symmetry. 6 faces. 2/m 3* 4* 3m Tetrahedron. Also isohedral, minimal and regular-faced. The truncated tetrahedron (4 triangles and 4 hexagons) is another example. 4 faces 432 Snub cube (6 squares, 32 triangles). Also regular-faced 23

Created 31 July 2001, Last Update 14 December 2009

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